1. Two Non Context-Free Languages (We will prove them later if we have time)

2. Regular languages Context-free languages Non Context-free languages

3. Positive Properties of Context-Free languages

4. Context-free languages are closed under: Union is context free is context free is context-free Union

5. Example Union Language Grammar

6. In general: The grammar of the union has new start variable and additional production For context-free languages with context-free grammars and start variables

7. Context-free languages are closed under: Concatenation is context free is context free is context-free Concatenation

8. In general: The grammar of the concatenation has new start variable and additional production For context-free languages with context-free grammars and start variables

9. Example Concatenation Language Grammar

10. Context-free languages are closed under: Star-operation is context free is context-free Star Operation

11. In general: The grammar of the star operation has new start variable and additional production For context-free language with context-free grammar and start variable

12. Example Language Grammar Star Operation

13. Context-free languages are closed under: reversal is context free is context-free Reversal

14. In general: The grammar of the reversal operation has production For context-free language with context-free grammar and every production

15. Example Language Grammar Reversal Operation

16. Negative Properties of Context-Free Languages

17. Context-free languages are not closed under: intersection is context free is context free not necessarily context-free Intersection

18. Example Context-free: Context-free: NOT context-free Intersection

19. Context-free languages are not closed under: complement is context free not necessarily context-free Complement

20. NOT context-free Example Context-free: Context-free: Complement

21. Is the complement of context free? Hw #1 p.219 For n a ( w ) > n b ( w ) : | cS 1 | S 1 c | cE | Ec This language is not context free (ref. Ex.8.8 p.217), but its complement is CF.

22. Intersection of Context-free languages and Regular Languages

23. The intersection of a context-free language and a regular language is a context-free language context free regular context-free

24. for for NPDA DFA Construct a new NPDA machine that accepts Machine Machine context-free regular simulates in parallel and i.e. whenever a symbol is read from input, M simultaneously executes the moves of M 1 and M 2

25. transition transition NPDA DFA transition NPDA

26. initial state initial state NPDA DFA Initial state NPDA

27. final state final states NPDA DFA final states NPDA

28. simulates in parallel and accepts string if and only if accepts string and accepts string

29. Therefore: (since is NPDA) is context-free is context-free

30. Applications of Regular Closure HW # 1, 7, 12, 17, 18, 21 ( 8.2 p.220)

31. The intersection of a context-free language and a regular language is a context-free language context free regular context-free Regular Closure

32. An Application of Regular Closure Prove that: is context-free

33. We know: is context-free

34. is regular is regular We also know:

35. regular context-free is context-free is context-free (regular closure)

36. Another Application of Regular Closure Prove that: is not context-free

37. context-free regular context-free If is context-free Then Impossible!!! (regular closure) Therefore, is not context free

38. Consider set difference: L 1 - L 2 Is it closed for context free languages? It is context free if L 2 is regular.